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Number of form 2^(2^p-1)-1 with prime exponent
In mathematics, a double Mersenne number is a Mersenne number of the form
where p is prime.
The first four terms of the sequence of double Mersenne numbers are[1] (sequence A077586 in the OEIS):
A double Mersenne number that is prime is called a double Mersenne prime. Since a Mersenne number Mp can be prime only if p is prime, (see Mersenne prime for a proof), a double Mersenne number M M p {\displaystyle M_{M_{p}}} can be prime only if Mp is itself a Mersenne prime. For the first values of p for which Mp is prime, M M p {\displaystyle M_{M_{p}}} is known to be prime for p = 2, 3, 5, 7 while explicit factors of M M p {\displaystyle M_{M_{p}}} have been found for p = 13, 17, 19, and mersenne prime 31.
Thus, the smallest candidate for the next double Mersenne prime is M M 61 {\displaystyle M_{M_{61}}} , or 22305843009213693951 − 1. Being approximately 1.695×10694127911065419641, this number is far too large for any currently known primality test. It has no prime factor below 1 × 1036.[2] There are probably no other double Mersenne primes than the four known.[1][3]
Smallest prime factor of M M p {\displaystyle M_{M_{p}}} (where p is the nth prime) are
The recursively defined sequence
is called the sequence of Catalan–Mersenne numbers.[4] The first terms of the sequence (sequence A007013 in the OEIS) are:
Catalan discovered this sequence after the discovery of the primality of M 127 = c 4 {\displaystyle M_{127}=c_{4}} by Lucas in 1876.[1][5][6]p. 22 Catalan conjectured that they are prime "up to a certain limit". Although the first five terms are prime, no known methods can prove that any further terms are prime (in any reasonable time) simply because they are too huge. However, if c 5 {\displaystyle c_{5}} is not prime, there is a chance to discover this by computing c 5 {\displaystyle c_{5}} modulo some small prime p {\displaystyle p} (using recursive modular exponentiation). If the resulting residue is zero, p {\displaystyle p} represents a factor of c 5 {\displaystyle c_{5}} and thus would disprove its primality. Since c 5 {\displaystyle c_{5}} is a Mersenne number, such a prime factor p {\displaystyle p} would have to be of the form 2 k c 4 + 1 {\displaystyle 2kc_{4}+1} . Additionally, because 2 n − 1 {\displaystyle 2^{n}-1} is composite when n {\displaystyle n} is composite, the discovery of a composite term in the sequence would preclude the possibility of any further primes in the sequence.
If c 5 {\displaystyle c_{5}} were prime, it would also contradict the New Mersenne conjecture. It is known that 2 c 4 + 1 3 {\displaystyle {\frac {2^{c_{4}}+1}{3}}} is composite, with factor 886407410000361345663448535540258622490179142922169401 = 5209834514912200 c 4 + 1 {\displaystyle 886407410000361345663448535540258622490179142922169401=5209834514912200c_{4}+1} .[7]
In popular culture[edit]In the Futurama movie The Beast with a Billion Backs, the double Mersenne number M M 7 {\displaystyle M_{M_{7}}} is briefly seen in "an elementary proof of the Goldbach conjecture". In the movie, this number is known as a "Martian prime".
The footnote (indicated by the star) written by the editor Eugène Catalan, is as follows:Prouver que 261 − 1 et 2127 − 1 sont des nombres premiers. (É. L.) (*).
(*) Si l'on admet ces deux propositions, et si l'on observe que 22 − 1, 23 − 1, 27 − 1 sont aussi des nombres premiers, on a ce théorème empirique: Jusqu'à une certaine limite, si 2n − 1 est un nombre premier p, 2p − 1 est un nombre premier p', 2p' − 1 est un nombre premier p", etc. Cette proposition a quelque analogie avec le théorème suivant, énoncé par Fermat, et dont Euler a montré l'inexactitude: Si n est une puissance de 2, 2n + 1 est un nombre premier. (E. C.)
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